Contrast Production functions to Utility functions.

What common/ different

points can you identify?

Production Function Utility Function

Output from inputs Preferences level from purchases

Derived from technologies Derived from preferences

Marginal product of inputs Marginal utility of products

Isoquant Indifference Curve

Marginal Rate of Technical Substitution Marginal Rate of Substitution

 Consider the production function whose equation is the formula 𝑸 = 𝑲𝟏/𝟐 𝑳𝟏/𝟐

1) Find the production function in table form, 2) Find mathematically the marginal as
given the following combinations of labor and well as the average products of both
capital. Which (L,K) combinations belong to the labor and capital.
same isoquant? Deduce the 𝑴𝑹𝑻𝑺𝑳,𝑲 and 𝑴𝑹𝑻𝑺𝑲,𝑳

K we have 𝑄 = 𝐾1/2𝐿1/2
0 10 20 30 40 50
L therefore :
∆𝑄 𝟏 −𝟏/𝟐 𝟏/𝟐
0 0 0 0 0 0 0 𝑴𝑷𝑳 = = 𝑳 𝑲
∆𝐿 𝟐
∆𝑄 𝟏 −𝟏/𝟐 𝟏/𝟐
10 0 10 14 17 20 22 𝑴𝑷𝑲 = = 𝑲 𝑳
∆𝐾 𝟐
𝑄 𝐾1/2 𝐿1/2
20 0 14 20 24 28 32 𝑨𝑷𝑳 = = = 𝑲𝟏/𝟐 𝑳−𝟏/𝟐
𝐿 𝐿
𝑄 𝐾1/2 𝐿1/2
30 0 17 24 30 35 39 𝑨𝑷𝑲 = = = 𝑲−𝟏/𝟐 𝑳𝟏/𝟐
𝐾 𝐾
40 0 20 28 35 40 45 1 −1/2 1/2
𝑀𝑃𝐿 2 𝐿 𝐾 𝑲
𝑴𝑹𝑻𝑺𝑳,𝑲 = = =
𝑀𝑃𝐾 1 𝐾 −1/2𝐿1/2 𝑳
50 0 22 32 39 45 50 2

1 −1/2 1/2
𝑀𝑃𝐾 2 𝐾 𝐿 𝑳
𝑴𝑹𝑻𝑺𝑲,𝑳 = = =
𝑀𝑃𝐿 1 𝐿−1/2𝐾1/2 𝑲
2
 4) Prove that the elasticity of
3) What is the equation of the isoquant
substitution for this production function
for 𝑸 = 𝟐𝟎 ?
is 𝝈 = 𝟏
Verify that the combinations identified
How does this indicate the shape of the
earlier are consistent with the equation.
corresponding isoquants ?
if 20 = 𝐾1/2 𝐿1/2 we know that 𝐾
then : %∆ 𝐿
𝜎 =
% ∆𝑀𝑅𝑇𝑆𝑙,𝐾
202 = (𝐾1/2𝐿1/2 )2

𝐾
400 = 𝐾 𝐿 and we have 𝑀𝑅𝑇𝑆𝐿,𝐾 =
𝐿
𝟒𝟎𝟎 𝐾
𝑲= therefore ∆ 𝐿
𝑳 ൙𝐾

𝝈 = 𝐿 =𝟏
𝐾
∆ 𝐿
൙𝐾
𝐿
the isoquants are curves.
5) Does this production function exhibit
decreasing, constant, or diminishing
returns to scale? Prove your anwser
mathematically.

we know that RTS is a measure of how

much the output will increase when ALL
inputs increase by a particular amount

let's suppose that initially:

𝑄1 = 𝐾11/2 𝐿11/2
we now increase both factors by 𝒙 to
obtain a new quantity:
𝑄2 = (𝑥𝐾1 )1/2 (𝑥𝐿1 )1/2

1/2 1/2
𝑄2 = 𝑥1/2 𝐾1 𝑥1/2 𝐿1

𝑄2 = 𝑥 (𝐾11/2 𝐿11/2)

𝑸𝟐 = 𝒙 𝑸𝟏

the output quantity increased at the

same proportion as the inputs,
therefore this production function
exhibits constant returns to scale.
The table below shows the total product function for a
glove knitting line at a local clothes factory where:
L represents the number of workers;
Q represents the total number of gloves knitted per
day.
1) Calculate the average and marginal products per
worker per day.
2) Plot the respective graphs and interpret them.
AP(L) = MP(L) =
L Q = TP(L)
TP(L)/L ∆TP(L)/∆L
0 0 - -
1 5 5,00 5
2 15 7,50 10
3 28 9,33 13
4 41 10,25 13
5 52 10,40 11
6 60 10,00 8
7 65 9,29 5
8 67 8,38 2
9 68 7,56 1
10 68,5 6,85 0,5
11 68,5 6,23 0
12 67 5,58 -1,5
 Q = TP(L)
80
(number of gloves per day)

70
60
Total Product

50
40
30
20
10
0
0 1 2 3 4 5 6 7 8 9 10 11 12
Labor (number of workers per day)

AP(L) and MP(L)

(number of gloves per worker per day)

14
Average and Marginal Products

12
10
8
6
4
2
0
-2 0 1 2 3 4 5 6 7 8 9 10 11 12

Labor (number of workers per day)

AP(L) = TP(L)/L MP(L) = ∆TP(L)/∆L

Notice that :

*When the number of workers is small, there is an increasing marginal return to labor (MP(L) is
increasing), but after 4 workers, there is a deceasing marginal returns to labor (MP(L) is decreasing).

*When few workers are employed, they may help each other perform tasks as a team, so an additional
worker may contribute to the overall productivity more than the previous additional worker.

*However, with fixed capital, as the number of workers increases, at some point the contribution of an
additional worker to output will start decreasing.

*The output is at its maximum when marginal product is equal to zero, that is by the addition of the 11th
worker.

*Therefore, starting from the 12th worker, the total output falls and there is diminishing total returns.

*As long as the marginal is above the average, the average is increasing, and whenever the marginal is
below the average, the average is decreasing.